Chapter 7.3, Problem 84E

Solution

To find: The f so that its graph contains the given points. It is given that $f\left(x\right)=a{x}^3+b{x}^2+cx+d$ and passes through $\left(-2,-37\right),\left(-1,-11\right),\left(0,-5\right),\left(2,19\right)$ .

The function is $f\left(x\right)=3x^3-x^2+2x-5$ .

Given information:

It is given that $f\left(x\right)=a{x}^3+b{x}^2+cx+d$ and passes through $\left(-2,-37\right),\left(-1,-11\right),\left(0,-5\right),\left(2,19\right)$ .

Explanation:

Consider the given information.

Use the given points and setup the system of equation.

For $\left(-2,-37\right)$ .

  $\begin{array}{c}-37=a{\left(-2\right)}^3+b{\left(-2\right)}^2+c\left(-2\right)+d\\ -8a+4b-2c+d=-37\end{array}$

For $\left(-1,-11\right)$ .

  $\begin{array}{c}-11=a{\left(-1\right)}^3+b{\left(-1\right)}^2+c\left(-1\right)+d\\ -a+b-c+d=-11\end{array}$

For $\left(0,-5\right)$ .

  $\begin{array}{c}-5=a{\left(0\right)}^3+b{\left(0\right)}^2+c\left(0\right)+d\\ d=-5\end{array}$

For $\left(2,19\right)$ .

  $\begin{array}{c}19=a{\left(2\right)}^3+b{\left(2\right)}^2+c\left(2\right)+d\\ 8a+4b+2c+d=19\end{array}$

Now solve the equation $AX=B$ where, $A=\left[\begin{array}{cccc}-8& 4& -2& 1\\ -1& 1& -1& 1\\ 0& 0& 0& -5\\ 8& 4& 2& 1\end{array}\right],X=\left[\begin{array}{c}a\\ b\\ c\\ d\end{array}\right]\text{ and }B=\left[\begin{array}{c}-37\\ -11\\ -5\\ 19\end{array}\right]$

The equation can be written as $X=A^{-1}B$ now solve by using the calculator.

  

Hence, $a=4,b=-1,c=2\text{ and }d=-5$ . Thus, the function is $f\left(x\right)=3x^3-x^2+2x-5$ .